Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $k \neq 0$. $t = \dfrac{k + 7}{k - 1} \times \dfrac{k^2 - 10k + 9}{k + 7} $
Solution: First factor the quadratic. $t = \dfrac{k + 7}{k - 1} \times \dfrac{(k - 1)(k - 9)}{k + 7} $ Then multiply the two numerators and multiply the two denominators. $t = \dfrac{ (k + 7) \times (k - 1)(k - 9) } { (k - 1) \times (k + 7) } $ $t = \dfrac{ (k + 7)(k - 1)(k - 9)}{ (k - 1)(k + 7)} $ Notice that $(k + 7)$ and $(k - 1)$ appear in both the numerator and denominator so we can cancel them. $t = \dfrac{ \cancel{(k + 7)}(k - 1)(k - 9)}{ \cancel{(k - 1)}(k + 7)} $ We are dividing by $k - 1$ , so $k - 1 \neq 0$ Therefore, $k \neq 1$ $t = \dfrac{ \cancel{(k + 7)}\cancel{(k - 1)}(k - 9)}{ \cancel{(k - 1)}\cancel{(k + 7)}} $ We are dividing by $k + 7$ , so $k + 7 \neq 0$ Therefore, $k \neq -7$ $t = k - 9 ; \space k \neq 1 ; \space k \neq -7 $